Analysis of a time optimal control problem related to the management of a bioreactor

DOI:10.1051/cocv/2010020 www.esaim-cocv.org

ANALYSIS OF A TIME OPTIMAL CONTROL PROBLEM RELATED TO THE MANAGEMENT OF A BIOREACTOR

Lino J. Alvarez-V´ azquez

, Francisco J. Fern´ andez

and Aurea Mart´ ınez

Abstract. We consider a time optimal control problem arisen from the optimal management of a bioreactor devoted to the treatment of eutrophicated water. We formulate this realistic problem as a state-control constrained time optimal control problem. After analyzing the state system (a complex system of coupled partial differential equations with non-smooth coefficients for advection-diffusion- reaction with Michaelis-Menten kinetics, modelling the eutrophication processes) we demonstrate the existence of, at least, an optimal solution. Then we present a detailed derivation of a first order opti- mality condition (involving the corresponding adjoint systems) characterizing these optimal solutions.

Finally, a numerical example is shown.

Mathematics Subject Classification.35D05, 49J20, 93C20.

Received February 10, 2009. Revised September 3, 2009 and January 21, 2010.

Published online April 23, 2010.

1. Introduction: Environment and mathematics

An excessive concentration of nutrients (usually nitrogen or phosphorus) in large bodies of water (as lakes, estuaries, rivers and so on) encourages the growth of aquatic organisms, causing its aging by an abnormal biological enrichment of the water. For instance, in young lakes water is clear and cold, supporting a reduced quantity of life. However, urban sewage and agricultural/industrial wastes derived from human activities can introduce into the aquatic media a large amount of nutrients which promotes an increasing of the lake’s fertility and accelerates the aging process: plant/animal life explodes, and organic detritus begin to be deposited on the bottom, giving raise to the eutrophication processes. The main pollutants (nitrates and phosphates) from drained wastewater act as plant nutrients, stimulating the algal blooms and, consequently, robbing the water of dissolved oxygen vital to other aquatic life species, and warming the water. These facts usually poison whole populations of fish, whose organic remains further exhaust the dissolved oxygen from water, leading the lake to ecological death.

Keywords and phrases. Time optimal control, partial differential equations, optimality conditions, existence, bioreactor.

∗Supported by Project MTM2009-07749 of Micinn (Spain), and INCITE-09-291-083-PR of Xunta de Galicia.

∗∗ The authors thank the fruitful suggestions of Prof. Jean-Pierre Raymond. The authors also appreciate the interesting and detailed remarks of the anonymous referee.

1 Departamento de Matem´atica Aplicada II, E.T.S.I. Telecomunicaci´on, Universidad de Vigo, 36310 Vigo, Spain.

lino@dma.uvigo.es; aurea@dma.uvigo.es

2 Departamento de Matem´atica Aplicada, Facultad de Matem´aticas, Universidad de Santiago de Compostela, 15706 Santiago, Spain. fjavier.fernandez@usc.es

Article published by EDP Sciences c EDP Sciences, SMAI 2010

Eutrophication of aquatic media has been considered one of the major threats to the health of ecosystems since the last decades. Eutrophication processes have been the subject of a wide range of biological/engineering researches but, from a purely mathematical viewpoint, related mathematical models have been much less an- alyzed. There exist several interesting papers studying the problem from a simplified ordinary differential equations viewpoint, but within the framework of partial differential equations, only a few models have been proposed: a 1D spatial model for oxygen dynamics was initially given by Lunardini and Di Cola [13]. More complex 2D depth-averaged models for nutrient-phytoplankton in shallow water have been given, for instance, by Arinoet al.[5] or Cioffi and Gallerano [9]. For the full 3D case, several numerical models have been proposed, among others, by Drago et al. [10], Yamashiki et al. [20] or Park et al. [14], but only introducing numerical simulations for particular zones, without presenting theoretical results about existence or uniqueness of solution.

The most remarkable results of existence of solution for a complete nutrient-phytoplankton-zooplankton-oxygen model can be found in the paper of Allegretto et al. [1] (for a particular periodic case) or in the recent work of the authors [3] (presenting also uniqueness and regularity results for a realistic situation with non-smooth water velocity).

In order to reduce this harmful concentration of nutrients in water, one of the most useful techniques is related to the use of bioreactors. The operation of this type of bioreactors consists of holding up eutrophicated water (rich, for instance, in nitrogen) in a small number of large tanks where we add a certain (small) quantity of phytoplankton, that we let grow in order to absorb nitrogen from water. In the particular problem analyzed in this work we consider only two large shallow tanks with the same capacities (but possibly with different geometries). Water rich in nitrogen will fill the first tank Ω₁, where we will add a quantityρ¹of phytoplankton – which we will let freely grow for a permanence time T¹ – to drop nitrogen level down to a desired threshold.

Inside this first tank we are also interested in obtaining – with economical purposes – a certain quantity of organic detritus, since they are very estimated as agricultural fertilizers. Once reached the desired levels of nitrogen and organic detritus (settled in the bottom of the tank, then reclaimed for agricultural use), we will drain this first tank and pass water to the second tank Ω₂, where the same operation is repeated, by adding a new amount ρ² of phytoplankton. Water leaving this second fermentation tank after a permanence timeT² (thus, the total time for the whole process is T¹+T²) will be usually poor in nitrogen, but rich in detritus (again settled in the bottom) and phytoplankton (recovered from a final filtering). At this point, we are also interested – for economical and ecological reasons – in minimizing this final quantity of phytoplankton. Thus, the time optimal control problem will consist of finding the quantities (ρ¹, ρ²) of phytoplankton that we must add to each one of both tanks along the time, and the minimal permanence times (T¹, T²), so that the nitrogen levels be lower that maximum thresholds and detritus levels be higher that minimum thresholds, and in such a way that the final phytoplankton concentration be as reduced as possible.

From a mathematical point of view, this problem can be formulated as a time optimal control problem with state and control constraints, where the control variable (T¹, T², ρ¹, ρ²) is the vector of permanence times and phytoplankton quantities added in each tank, the state variables are the concentrations of nutrient, phytoplank- ton, zooplankton and organic detritus, the cost function to be minimized is a combination of the permanence times and the phytoplankton concentration of water leaving the second tank, the state constraints stand for the thresholds required for the final nitrogen and detritus concentrations in each one of the tanks, and the control constraints are related to technological bounds.

The mathematical literature related to the analysis of time optimal control problems is not large. However, the number of references devoted to time optimal control of partial differential equations is more reduced.

Among the most recent contributions on the topic we must mention those of Cannarsa and Frankowska [7]

and Gugat and Leugering [11], where a semigroup theory approach is used, or those of Raymond [4,15,16] and Wang [12,18,19], dealing with academical related problems with the time as a control variable. In these papers very interesting results on the existence of solutions and optimality conditions are obtained. Unfortunately, none of them can be applied to our particular problem due to the lack of smoothness in the coefficients, and to their dependence on time. So, we will need to develop a complete analysis of the time optimal control problem specifically adapted to our non-smooth assumptions.

A detailed mathematical formulation of the time optimal control problem is presented in Section 2. The next section is devoted to the analysis of the state system, and to the demonstration of the existence of a (non necessarily unique) optimal solution. Finally, these optimal solutions will be characterized by a first order optimality condition (involving a suitable adjoint system to be adequately defined) presented in Section 4. We will also introduce an alternative formulation of the optimality conditions for the qualified case.

2. Formulation of the time optimal control problem

Most realistic mathematical models governing eutrophication are recently formulated as systems of partial differential equations (as opposite to classical systems of ordinary differential equations) with a high complexity due to the great variety of internal phenomena appearing on them. In this work we have considered a complete model, where the four biological variables involved in our problem appear (the formulation of the biochemical interaction terms and their meaning can be found, for instance, in Canale [6]). Thus, we consider the state variableu= (u¹, u², u³, u⁴),whereu¹denotes a generic nutrient concentration, for instance, nitrogen (as will be considered in our case) or phosphorus,u²denotes the phytoplankton concentration,u³denotes the zooplankton concentration, andu⁴denotes the organic detritus concentration.

The interaction of these four biological species into a given water domain Ω⊂ R³ (with a smooth enough boundary∂Ω) and along a time intervalI= (0, T) can be described by the following system of coupled partial differential equations for convection-diffusion-reaction with standard Michaelis-Menten kinetics:

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

∂u¹

∂t +w· ∇u¹− ∇ ·(μ₁∇u¹) +C_ncL_K^u1

N+u¹u²−C_ncK_ru²−C_ncK_rdΘ^θ−20u⁴=g¹ inQ,

∂u²

∂t +w· ∇u²− ∇ ·(μ₂∇u²)−L_K^u1

N+u¹u²+K_ru²+K_mfu²+K_zK^u2

F+u²u³=g² inQ,

∂u³

∂t +w· ∇u³− ∇ ·(μ₃∇u³)−C_{f z}K_zK^u2

F+u²u³+K_mzu³=g³ inQ,

∂u⁴

∂t +w· ∇u⁴− ∇ ·(μ₄∇u⁴)−K_mfu²−K_mzu³+K_rdΘ^θ−20u⁴+W_{f d}^∂u_∂x⁴

3 =g⁴ inQ,

(2.1)

with corresponding boundary conditions on Σ and initial conditions in Ω, and where we have considered:

• Q=I×Ω, Σ =I×∂Ω;

• w(t, x): the water velocity;

• μ_i, i= 1, . . . ,4: the diffusion coefficients of each species, which are positive constants;

• C_nc: the nitrogen-carbon stoichiometric relation;

• L(t, x): the luminosity function, given by expression L(t, x) =νC_t^θ(t,x)−20I₀

I_se^{−φ x3},

withνthe maximum phytoplankton growth rate,C_tthe phytoplankton growth thermic constant,θ(t, x) the water temperature (in Celsius), I₀ the incident light intensity, I_s the light saturation, andφ the light absorption by water;

• K_N: the nitrogen half-saturation constant;

• K_rd: the detritus regeneration rate;

• Θ: the detritus regeneration thermic constant, which is positive;

• K_r: the phytoplankton endogenous respiration rate;

• K_mf: the phytoplankton death rate;

• K_z: the zooplankton predation (grazing);

• K_F: the phytoplankton half-saturation constant;

• C_{f z}: the grazing efficiency factor;

• K_mz: the zooplankton death rate (including predation);

• W_{f d}: the falling velocity of organic detritus (settling).

In order to present in a simple way the system of equations (2.1) we will consider, for R⁺ = [0,∞), the mapping A= (A¹, A², A³, A⁴) :R⁺×Ω×[R⁺]⁴−→R⁴, given by:

A(t, x,u) =

⎡

⎢⎢

⎢⎢

⎢⎢

⎣

−C_nc L(t, x)_K^u1

N+u¹u²−K_ru²

+C_ncK_rdΘ^θ(t,x)−20u⁴ L(t, x)_K^u1

N+u¹u²−K_ru²

−K_mfu²−K_zK^u2

F+u²u³ C_{f z}K_zK^u2

F+u²u³−K_mzu³ K_mfu²+K_mzu³−K_rdΘ^θ(t,x)−20u⁴

⎤

⎥⎥

⎥⎥

⎥⎥

⎦

. (2.2)

Moreover, the sedimentation term W_{f d}^∂u_∂x⁴

3 in the fourth equation of (2.1) will be incorporated into the convective termw· ∇u⁴, by redefining a new artificial velocityw+ (0,0, W_{f d}) for the organic detritus equation.

Thus, taking into account above considerations, the system (2.1) can be written in the following equivalent way:

∂uⁱ

∂t +w_i· ∇uⁱ− ∇ ·(μ_i∇uⁱ) =Aⁱ(t, x,u) +gⁱ inQ, for i= 1, . . . ,4, (2.3) wherew_i=wfori= 1,2,3,andw₄=w+ (0,0, W_{f d}).

With these notations we can mathematically formulate the time optimal control problem with the following elements:

• Controls: As we have remarked in the Introduction, we will control the system by means of four design variables: the permanence time T¹ of water inside the first tank Ω₁, the permanence time T² inside the second tank Ω₂, the quantityρ¹(t, x) of phytoplankton added in the first tank along the time interval I₁ = (0, T¹), and the quantity ρ²(t, x) of phytoplankton added in the second tank along the corresponding time intervalI₂= (0, T²).

It is worthwhile remarking here that, although the time-space variables are different for each tank ((t₁, x₁)∈I₁×Ω₁for the first tank, (t₂, x₂)∈I₂×Ω₂for the second one), for the sake of simplicity, we will use the same variables (t, x) for both cases, since no confusion is possible.

• State systems: We consider two state systems giving the concentrations of nitrogen, phytoplankton, zooplankton, and organic detritus in each one of the tanks. Since both tanks are isolated, no transference for any of the four species is considered across their boundaries, that is, Neumann boundary conditions are assumed to be null for all concentrations in both tanks. In addition, both systems are coupled by means of the initial-final conditions: when water passes from the first tank to the second one, it is natural to assume that water is mixed up, and this is the reason of considering the initial conditions for the concentrations inside the second tank as given by the corresponding averaged final concentrations in the first tank. So, the two coupled state systems are given by:

– TankΩ₁: The state variables for the first tank will be denotedu¹= (u^1,1, u^2,1, u^3,1, u^4,1) withu^1,1 (nitrogen),u^2,1(phytoplankton),u^3,1(zooplankton), andu^4,1(organic detritus). The permanence time of water inside this first tank will be T¹, the fluid velocity in the tank will be w¹(t, x), and the initial concentrations will be given byu¹₀= (u^1,1₀ , u^2,1₀ , u^3,1₀ , u^4,1₀ ). Thus, for Q₁=I₁×Ω₁ and Σ₁=I₁×∂Ω₁, we have the system, fori= 1, . . . ,4:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∂u^i,1

∂t +w¹_i · ∇u^i,1− ∇ ·(μ_i∇u^i,1) =Aⁱ(t, x,u¹) +

ρ¹, if i= 2

0, if i= 2 in Q₁,

∂u^i,1

∂n = 0 on Σ₁,

u^i,1(0) =u^i,1₀ in Ω₁.

(2.4)

– Tank Ω₂: The state variables for the second tank will be denotedu²= (u^1,2, u^2,2, u^3,2, u^4,2) with u^1,2(nitrogen),u^2,2(phytoplankton),u^3,2 (zooplankton), andu^4,2(organic detritus). The perma- nence time of water inside this second tank will beT², and the fluid velocity in the tank will be

w²(t, x). Thus, forQ₂=I₂×Ω2and Σ₂=I₂×∂Ω₂, we have the analogous system, fori= 1, . . . ,4:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∂u^i,2

∂t +w²_i · ∇u^i,2− ∇ ·(μ_i∇u^i,2) =Aⁱ(t, x,u²) +

ρ², if i= 2

0, if i= 2 inQ₂,

∂u^i,2

∂n = 0 on Σ₂,

u^i,2(0) = _meas(Ω¹

1)M₁ⁱ(u¹(T¹)) in Ω₂,

(2.5)

where M_j = (M_j¹, M_j², M_j³, M_j⁴), for j = 1,2, are the functionals, defined from [L¹(Ω_j)]⁴ to R⁴, given by:

M_j(v^j) =

⎡

⎢⎢

⎢⎢

⎢⎣

Ωjv^1,jdx

Ωjv^2,jdx

Ωjv^3,jdx 0

⎤

⎥⎥

⎥⎥

⎥⎦

. (2.6)

We have to note here that, since detritus settle before water passes to the second tank, the ini- tial organic detritus concentrationu^4,2(0) (that is, the fourth component of M₁(u¹(T¹))) will be considered null.

• Objective function: Due to the fact that we are interested in reducing the total processing time T¹+T² of water inside the bioreactor, and in minimizing the final phytoplankton concentration of water leaving the second tank, we are led to consider the following cost functionalJ given by:

J(T¹, T², ρ¹, ρ²) =N₁(T¹+T²) + N₂ meas(Ω₂)

Ω2

u^2,2(T²)dx, (2.7)

whereN₁, N₂≥0 are two scaling factors.

• State constraints: As commented in the first section, the final nitrogen concentration in each tank must be lower than a given threshold, and the final organic detritus concentration in each tank must be greater than another given threshold. These constraints translate into the relations given by B= (B¹, B², B³, B⁴):

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

B¹(T¹, T², ρ¹, ρ²) = _meas(Ω¹

1)

Ω1u^1,1(T¹)dx≤σ₁, B²(T¹, T², ρ¹, ρ²) = _meas(Ω¹

2)

Ω2u^1,2(T²)dx≤σ₂, B³(T¹, T², ρ¹, ρ²) = _meas(Ω¹

1)

Ω1u^4,1(T¹)dx≥θ₁, B⁴(T¹, T², ρ¹, ρ²) = _meas(Ω¹

2)

Ω2u^4,2(T²)dx≥θ₂,

(2.8)

for certain given valuesσ₁, σ₂, θ₁, θ₂>0.

• Control constraints: Finally, for technological reasons, the permanence times (T¹, T²) must take their values between two fixed bounds 0< T_min< T_max<∞, that is, they must lie in the set

V_ad¹ ={(T¹, T²)∈R²:T_min≤T¹, T²≤T_max} (2.9) which is a compact, convex, and nonempty subset ofR².

By similar motivations, the quantities (ρ¹, ρ²) of phytoplankton added to the tanks must be nonneg- ative and bounded by a maximal admissible valueK >0, that is, they must lie in the set

V_ad² = {(ρ¹, ρ²)∈L²((0, T_max)×Ω₁)×L²((0, T_max)×Ω₂) : 0≤ρ¹(t, x)≤K a.e. (t, x)∈(0, T_max)×Ω₁, 0≤ρ²(t, x)≤K a.e. (t, x)∈(0, T_max)×Ω₂}

(2.10)

which is a closed, bounded, convex, and nonempty subset ofL²((0, T_max)×Ω₁)×L²((0, T_max)×Ω₂).

In order to simplify the notations, we will denote Vad = V_ad¹ × V_ad² . So, the control must satisfy (T¹, T², ρ¹, ρ²)∈ Vad.

Thus, the formulation of the time optimal control problem, denoted by (P), will be the following:

(P) inf

J(T¹, T², ρ¹, ρ²) such that (T¹, T², ρ¹, ρ²)∈ Vad and (u¹,u²) satisfies (2.4)–(2.5) and (2.8) .

3. Analysis of the control problem

Since the problem we are dealing with depends on varying (although bounded) final times, we need to assume that the water velocities w^j, j = 1,2, and the corresponding water temperatures θ^j, j = 1,2, in the coupled state systems (2.4)–(2.5) are defined all along the whole time interval [0, T_max].

As a first step we will introduce several notations for the functional spaces to be used in the below sections:

We denote L³_σ(Ω_j) the closure in L³(Ω_j)³ of the space {v ∈ D(Ω_j)³ : div(v) = 0}, j = 1,2. We define V_j = [H¹(Ω_j)]⁴, H_j= [L²(Ω_j)]⁴,j= 1,2, which can be considered within the classical frameworkV_j ⊂H_j≡ H_j⊂V_j,j= 1,2. Finally,W^1,p,q(0, T_max;V, V) denotes the functional space given by:

W^1,p,q(0, T_max;V, W) =

v ∈L^p(0, T_max;V) : dv

dt ∈L^q(0, T_max;W)

,

for any Banach spacesV ⊂W, and for 1≤p, q≤ ∞.

Then, as was demonstrated by the authors in [3] (using similar arguments to previous paper [2] for a problem related to food technology), the eutrophication systems (2.4)–(2.5) admit a solution under non-smooth hypothe- ses. To be exact, if we assume that the initial conditions and the source terms are nonnegative and bounded, and that the fluid velocities and temperatures satisfy:

w^j ∈L^2+δ(0, T_max;L³_σ(Ω)), j= 1,2, for some δ≥0, θ^j∈L²((0, T_max)×Ω_j), j= 1,2,

such that 0≤θ_j(t, x)≤M a.e. (t, x)∈(0, T_max)×Ω_j,

(3.1)

then the state systems (2.4)–(2.5) admit a unique solutionu^j∈W^1,2,1+(0, T_max;V_j,V_j)∩[L^∞((0, T_max)×Ωj)]⁴, j= 1,2, for a certain small ≥0. (In particular, forδ= 0 we have = 0, and forδ→ ∞, we have →1; see [3].) Moreover, this solution (u¹,u²) is nonnegative and bounded (in previous space norm) by a value depending on the time boundT_max, the fluid velocities (w¹,w²), the controls (ρ¹, ρ²), and the initial conditionu¹₀.

Unfortunately these mild hypotheses – which will assure the existence of solution for the time optimal control problem (P) – are not enough in order to allow us the derivation of optimality conditions for this problem. To do this we will need to impose the following stronger hypotheses on water velocities and temperatures:

w^j ∈W^1,∞,∞(0, T_max;L³_σ(Ω_j), L³_σ(Ω_j)), j= 1,2, θ^j∈W^1,∞,∞(0, T_max;L^∞(Ω_j), L^∞(Ω_j)), j= 1,2,

such that 0≤θ_j(t, x)≤M a.e. (t, x)∈(0, T_max)×Ω_j.

(3.2)

Finally, we will introduce the following definition:

Definition 3.1. We will say that (T¹, T², ρ¹, ρ²)∈ Vad is a feasible control for the time optimal control prob- lem (P) if the associated state (u¹,u²), solution of coupled systems (2.4)–(2.5), satisfies the constraints (2.8).

Then, we can prove the following existence result under mild hypotheses (3.1):

Theorem 3.2. Let us assume that the set of feasible controls is nonempty. Let u¹₀ ∈ [L^∞(Ω₁)]⁴ be such that 0≤u^i,1₀ (x)≤M a.e. x∈Ω₁, i= 1, . . . ,4, and let us assume that the water velocities and temperatures (w¹,w², θ¹, θ²)satisfy the regularity hypotheses(3.1), forδ >0. Then, there exist elements(T¹,T²,ρ¹,ρ²,u¹,u²)

∈ Vad×(W^1,2,1+(0,T¹;V₁,V₁)∩[L^∞((0,T¹)×Ω₁)]⁴)×(W^1,2,1+(0,T²;V₂,V₂)∩[L^∞((0,T²)×Ω₂)]⁴), with > 0, such that (T¹,T²,ρ¹,ρ²) is a solution of the time optimal control problem (P) with associated state (u¹,u²).

Proof. To prove the existence of optimal control for problem (P) we will consider a minimizing sequence {(T_n¹, T_n², ρ¹_n, ρ²_n,u¹_n,u²_n)}n∈N⊂ Vad×(W^1,2,1+(0, T_n¹;V₁,V₁)∩[L^∞((0, T_n¹)×Ω1)]⁴)×(W^1,2,1+(0, T_n²;V₂,V₂)∩

[L^∞((0, T_n²)×Ω2)]⁴), that is, for alln∈N, the pair (u¹_n,u²_n) verifies the state systems (2.4)–(2.5) with associated control (ρ¹_n, ρ²_n), satisfies the constraints (2.8), and:

n→∞lim J(T_n¹, T_n², ρ¹_n, ρ²_n) = inf

J(T¹, T², ρ¹, ρ²) such that (T¹, T², ρ¹, ρ²)∈ Vad

and (u¹,u²) satisfies (2.4)–(2.5) and (2.8) .

From the uniqueness of solution of state systems (2.4)–(2.5) we can assure that (u¹_n,u²_n) are defined for the whole time interval (0, T_max), for alln∈N. Then we have, thanks to the estimates obtained in [3] and to the boundedness of the setVad, that the sequence of states{(u¹_n,u²_n)}_n∈N is bounded,i.e.

u^j_nW^1,2,1+_(0,Tmax;Vj,V

j)∩[L^∞((0,Tmax)×Ωj)]⁴ ≤C(T_max, M), j= 1,2, ∀n∈N. (3.3) Thus, since the inclusionW^1,2,1+(0, T_max;V_j,V_j)∩[L^∞((0, T_max)×Ωj)]⁴⊂[L^103−γ((0, T_max)×Ωj)]⁴is compact

∀γ >0 (for instance, applying Lem. 7.8 of Roub´ıˇcek [17] with the choicesV₁=H¹(Ω_j), V₂=L^6−δ(Ω_j) (for any δ >0 small we have that the inclusionH¹(Ω_j)⊂L^6−δ(Ω_j) is compact),V₃ =H¹(Ω_j), H =L²(Ω_j) andV₄= L^μ(Ω_j) (with any μ⁻¹= ^1−λ₂ +_6−δ^λ for λ∈(0,1)), we have the compact inclusion W^1,2,1+(0, T_max;V_j,V_j)∩ L^∞((0, T_max)×Ω_j)⊂L^λ2(0, T_max;L^μ(Ω_j)); then, by choosingδ >0 such thatμ= ²_λ = ¹⁰₃ −γ we obtain the desired inclusion (we must recall here that p= ¹⁰₃ is the greatest exponent psuch that L²(0, T_max;H¹(Ω_j))∩ L^∞(0, T_max;L²(Ω_j))⊂L^p((0, T_max)×Ω_j)), there exist subsequences, still denoted in the same way, such that:

ρ^j_n ρ^j inL²((0, T_max)×Ω_j), j= 1,2, u^j_n u^j inL²(0, T_max;V_j), j= 1,2,

u^j_n → u^j in [L¹⁰3−γ((0, T_max)×Ω_j)]⁴, ∀γ >0, j= 1,2, u^j_n(t) u^j(t) inH_j, ∀t∈[0, T_max], j= 1,2,

T_n^j → T^j inR, j= 1,2.

(3.4)

As a first point we will demonstrate that (u¹,u²) is the state associated to the control (T¹,T²,ρ¹,ρ²).

In order to do this, we will pass to the limit in the state systems satisfied by the states (u¹_n,u²_n) associated to the controls (T_n¹, T_n², ρ¹_n, ρ²_n). We consider the very weak formulation of these state systems as defined

in (0, T_max)×Ω₁ and (0, T_max)×Ω₂, respectively:

4 i=1

⎧⎨

⎩

(0,Tmax)×Ω1

μ_i∇u^i,1_n · ∇zⁱdxdt+

(0,Tmax)×Ω1

w¹_i · ∇u^i,1_n zⁱdxdt− _Tmax

0

dzⁱ

dt (t), u^i,1_n (t)

V₁,V1

dt

⎫⎬

⎭

= 4 i=1

(0,Tmax)×Ω1

Aⁱ(t, x,u¹_n)zⁱdxdt+

Ω1

u^i,1₀ zⁱ(0)dx

+

(0,Tmax)×Ω1

ρ¹_nz²dxdt,

∀z∈W^1,∞,∞(0, T_max;V₁,V₁) such that z(T_max) = 0, (3.5) 4

i=1

⎧⎨

⎩

(0,Tmax)×Ω2

μ_i∇u^i,2_n · ∇zⁱdxdt+

(0,Tmax)×Ω2

w²_i · ∇u^i,2_n zⁱdxdt− _Tmax

0

dzⁱ

dt (t), u^i,2_n (t)

V₂,V2

dt

⎫⎬

⎭

= 4 i=1

(0,Tmax)×Ω2

Aⁱ(t, x,u²_n)zⁱdxdt+ 1 meas(Ω₁)

3 i=1

Ω2

Ω1

u^i,1_n (T_n¹)dx

zⁱ(0)dx+

(0,Tmax)×Ω2

ρ²_nz²dxdt,

∀z∈W^1,∞,∞(0, T_max;V₂,V₂) such that z(T_max) = 0. (3.6) Passing to the limit in all the terms in (3.5) and in almost all the terms in (3.6) is a direct consequence of the convergences in (3.4). The only terms in (3.6) where we can not pass to the limit in an immediate way are those of the form

Ω1u^i,1_n (T_n¹)dx. However, for those terms the pass to the limit can be obtained by means of the change of variable τ = _T^t1

n in the state system corresponding to tank 1. So, if we denote v_n^i,1(τ) =u^i,1_n (T_n¹τ) =u^i,1_n (t), this state system turns into:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∂v^i,1_n

∂τ +T_n¹w¹_i · ∇v_n^i,1−T_n¹∇ ·(μ_i∇v^i,1_n ) =T_n¹Aⁱ(t, x,v¹_n) +δ_2iT_n¹ρ¹_n in (0,1)×Ω₁,

∂v^i,1_n

∂n = 0 on (0,1)×∂Ω₁,

v_n^i,1(0) =u^i,1₀ in Ω₁,

whereδ_2i denotes the Kronecker’s delta, that is,δ_2i = 1 ifi= 2, andδ_2i= 0 ifi= 2.

Arguing as for equation (3.5) we can pass to the limit in above system, and obtain the existence of a subsequence of{v¹_n}n∈N (still denoted in the same way) converging tov¹, withv^i,1(τ) =u^i,1(T¹τ), in similar spaces as{u¹_n}n∈Nin (3.4). In particular, we have that:

v_n¹(τ)v¹(τ) inH₁, ∀τ ∈[0,1]. (3.7) But, this convergence implies, in a direct manner, that:

n→∞lim

Ω1

v^i,1_n (1)dx=

Ω1

v^i,1(1)dx, (3.8)

or, equivalently, that:

n→∞lim

Ω1

u^i,1_n (T_n¹)dx=

Ω1

u^i,1(T¹)dx. (3.9)

So, we have just proved that (u¹,u²) is the unique state associated to the control (T¹,T²,ρ¹,ρ²).

On the other hand, from the properties of Vad, it is clear that (T¹,T²,ρ¹,ρ²) ∈ Vad. Thus, we are only left to prove that (u¹,u²) satisfy the constraints (2.8). The first and third constraints are satisfied, as a direct consequence of convergence (3.9). For demonstrating the verification of the other two constraints we only need

to use the new change of variable τ = _T^t2

n in the state system corresponding to Tank 2. So, if we denote v_n^i,2(τ) =u^i,2_n (T_n²τ) =u^i,t_n (t), by similar arguments to those given for the state system corresponding to Tank 1, we obtain that:

n→∞lim

Ω2

u^i,2_n (T_n²)dx=

Ω2

u^i,2(T²)dx, (3.10)

from which we conclude that constraints (2.8) are satisfied.

Moreover, by convergences (3.4) and (3.10), we deduce that:

n→∞lim J(T_n¹, T_n², ρ¹_n, ρ²_n) =J(T¹,T²,ρ¹,ρ²), (3.11) i.e. (T¹,T²,ρ¹,ρ²) is a solution of the time optimal control problem (P).

4. Optimality conditions

In this section we will obtain the necessary first order optimality conditions for characterizing the solutions of the time optimal control problem (P). As noted in the previous section, since two of the control variables are the permanence times (T¹, T²), we will need to guarantee the differentiability of the states with respect to the timet. So, we will need to impose the more restrictive hypotheses (3.2) on the fluid velocities and temperatures to assure the existence of optimality conditions. To obtain these optimality conditions we will use a change of time variable (introducing a new controlv in the spirit of the original ideas of Raymond and Zidani [15]), since a simple scaling of time has shown to be insufficient for our purposes. It is worthwhile remarking here that the time transformation by introducing a new controlv had been previously introduced – within the ordinary differential equations framework – in the classical book of Zeidler [21], for instance.

For a better organization and understanding of the results exposed in this section, we will divide it into two subsections. In the first one we will analyze how to obtain the optimality conditions for a new optimal control problem (P) with fixed final time, obtained from the problem (P) via a change of the time variable. In the second subsection we will use the results obtained in the first one in order to derive the optimality conditions for the original time optimal control problem (P).

4.1. The problem with fixed final time

For any given time interval [0, T] ⊂ R with T > 0, and for R⁺_∗ = (0,∞), we will consider a function v∈ C([0,1];R⁺_∗) such that:

₁

0

v(ζ)dζ=T. (4.1)

Associated to above function, we will define the following change of variable:

t=φ(τ) :=

_τ

0

v(ζ)dζ, τ∈[0,1], (4.2)

that implies ^dφ_dτ(τ) =v(τ), which will give rise to the new state equations.

Then if for each one of both tanks Ω_j,j= 1,2, we consider the corresponding change of variablet=φ^j(τ) = _τ

0 v^j(ξ)dξ, τ ∈[0,1], we will introduce the following notations:

y^j(τ) := (u^j◦φ^j)(τ) =u^j(t), ^j(τ) := (ρ^j◦φ^j)(τ) =ρ^j(t), j= 1,2. (4.3) Applying above changes of variable to our original optimal control problem with varying final time (P), we can rewrite it under the following alternative form of an optimal control problem with fixed final time:

• Controls: The four new design variables will be the functions v^j(τ),j = 1,2, for the changes of time variables, and the corresponding transformed quantities^j(τ, x) of phytoplankton added in the tank Ω_j, j= 1,2, along the new time interval (0,1).

• State systems: We consider the two transformed state systems for the concentrations of nitrogen, phytoplankton, zooplankton, and organic detritus in each one of the tanks, and two new state equations for the corresponding changes of time variable in each one of the tanks.

– Species in tankΩ₁:

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

∂y^i,1

∂τ +v¹(τ)w¹_i(φ¹(τ), x)· ∇y^i,1−v¹(τ)∇ ·(μ_i∇y^i,1)

=v¹(τ)Aⁱ(φ¹(τ), x,y¹) +δ_2iv¹(τ)¹(τ) in (0,1)×Ω₁,

∂y^i,1

∂n = 0 on (0,1)×∂Ω₁,

y^i,1(0) =u^i,1₀ in Ω₁.

(4.4)

– Species in tankΩ₂:

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

∂y^i,2

∂τ +v²(τ)w²_i(φ²(τ), x)· ∇y^i,1−v²(τ)∇ ·(μ_i∇y^i,2)

=v²(τ)Aⁱ(φ²(τ), x,y²) +δ_2iv²(τ)²(τ) in (0,1)×Ω₂,

∂y^i,2

∂n = 0 on (0,1)×∂Ω₂,

y^i,2(0) = _meas(Ω¹

1)Mⁱ₁(y¹(1)) in Ω₂.

(4.5)

– Change of time variable in tank Ω₁: dφ¹

dτ(τ) =v¹(τ) in (0,1),

φ¹(0) = 0. (4.6)

– Change of time variable in tank Ω₂: _dφ2

dτ(τ) =v²(τ) in (0,1),

φ²(0) = 0. (4.7)

• Objective function:

J(v¹, v², ¹, ²) =N_{1 1}

0

(v¹(τ) +v²(τ))dτ+ N₂ meas(Ω₂)

Ω2

y^2,2(1)dx. (4.8)

• State constraints:

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

B¹(v¹, v², ¹, ²) =_meas(Ω¹

1)

Ω1y^1,1(1)dx≤σ₁, B²(v¹, v², ¹, ²) =_meas(Ω¹

2)

Ω2y^1,2(1)dx≤σ₂ B³(v¹, v², ¹, ²) =_meas(Ω¹

1)

Ω1y^4,1(1)dx≥θ₁, B⁴(v¹, v², ¹, ²) =_meas(Ω¹

2)

Ω2y^4,2(1)dx≥θ₂.

(4.9)

• Control constraints: As commented in above paragraphs, functions (v¹, v²) corresponding to the changes of time variable must lie in the set

W_ad¹ ={(v¹, v²)∈ C([0,1];R⁺)² : T_min≤v¹(τ), v²(τ)≤T_max, ∀τ∈[0,1]}. (4.10)